# Basic Concepts in the Theory of Electrolytes

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## Thermodynamic Integration and Perturbation Theory

As mentioned before, the thermodynamic potentials cannot be directly obtained during a simulation (Molecular Dynamics or Monte Carlo), whereas their derivatives are easily measured. It is then natural to think of measuring the differences between the potentials by integrating their derivatives. For simplicity, we will focus on the canonical (NV T) ensemble, and thus on the Helmholtz free energy F(N; V; T). The generalization to other ensembles is straightforward.
Consider the two thermodynamic relations given in Eq. (3.20), which relate the derivatives of the free energy F with respect to volume V and temperature T, to the pressure P and the internal energy U, both of which can be directly measured during a simulation (See Equation (4.35)). The simplest procedure to obtain the free energy of a given (fluid) system, is to start at infinite dilution A = 0, and to integrate P with respect to , until the desired density B is reached. As the free energy of the ideal gas is known, we would thus obtain the absolute free energy, and not just a free energy difference. In general, this is only valid for low densities B, such that the thermodynamic trajectory does not cross a phase boundary. To overcome such a problem, a two-step thermodynamic integration can be performed, which avoids the coexistence region by starting at an artificial temperature TA above the critical temperature Tc. The system can then be compressed along the TA isotherm to the desired density B, before being cooled to the target temperature TB. A schematic representation of this procedure is given in Figure 3.1. Formally, the procedure is expressed as follows: we integrate the derivative of F with respect to (V), from A to B, to obtain AF(B; TA) N = AF(A; TA) N + Z B A d0 AP(0; TA) 02.

### Time Correlation Functions: The Green-Kubo Formalism

Up to this point, we have only discussed equilibrium properties, and no mention has been made of the dynamic properties, or how these can be obtained, as they fall into the domain of non-equilibrium statistical mechanics. If the central role in equilibrium statistical mechanics is played by the partition function, the corresponding role in the study of non-equilibrium phenomena is played by the time correlation function CAB(t), defined as an ensemble average by CAB(t) = hA(t)B(t = 0)i = Z d􀀀A(􀀀(t))B(􀀀(0))f(􀀀(0)).

#### Experimental Properties of Electrolytes Solutions

Before continuing with the theoretical description of electrolyte solutions, it is important to discuss how such systems are characterized in the laboratory. As most experimentsare performed at constant temperature T and pressure P, the natural thermodynamic potential is the Gibbs potential (Eq. (3.30)). For an electrolyte solution, the differential form of this potential is given by dG = V dP 􀀀 SdT + wdNw + X i idNi (3.77).where w and i refer to the water (solvent) and ion (solute) particles. Among the various thermodynamic properties of the system, the chemical potentials of the individual components w and i are probably the most important, since the Gibbs potential can be directly obtained from them. Experimentally, however, such properties are usually measured with respect to some standard state8, which we denote with a superscript (0). For the ions, the chemical potential difference i = i 􀀀 (0) i , is given directly by the ratio of the activities of the two states ai = zi=z(0) i , since kBT ln ai = i 􀀀 (0).

The MSA Solution

The problem with the DH theory is the fact that it does not consider the short-range interactions between particles, which become important as the concentration is increased.
The MSA solution (Eq. (3.51)) attempts to address these problems by taking into account the finite size of the particles, as well as the long-range electrostatic interactions. It has been widely adopted in the study of electrolytes, since explicit algebraic equations can be obtained for the excess (electrostatic) thermodynamic properties and the radial distribution functions. In order to obtain a complete description of electrolytes, within the primitive model, the results provided by the MSA theory must be complimented with the remaining (non-electrostatic) terms. We proceed to give a detailed description of the different contributions to the thermodynamics and the radial distribution functions of a primitive model, within the MSA description.

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1 Introduction et Résumé (version française)
1.1 Importance d’une description à plusieurs échelles
1.2 Plan de Travail
1.3 Résumé
2 Introduction
2.1 The Importance of a Multi-Scale Description
2.2 Plan of our Work
3 Basic Concepts in the Theory of Electrolytes
3.1 Statistical Thermodynamics of Simple Liquids
3.1.1 Statistical Averages and Distributions
3.1.2 Distribution Functions
3.1.3 Integral Equations
3.1.4 Thermodynamic Integration and Perturbation Theory
3.1.5 Time Correlation Functions: The Green-Kubo Formalism
3.2 Experimental Properties of Electrolytes Solutions
3.3 The Implicit Solvent Model
3.3.1 The Limiting Laws
3.3.2 The MSA Solution
3.3.3 The BIMSA Solution
3.4 Exact theories of Electrolyte Solutions
3.4.1 Introduction
3.4.2 Kirkwood-Buff Theory of Electrolyte Solutions
3.4.3 McMillan-Mayer Theory of Electrolyte Solutions
4 Ion-Specific Effects from Ab-Initio Descr.
4.1 Introduction
4.2 Principles of Ab-Initio Simulations
4.2.1 Solving the N-body Problem: A Variational Approach
4.2.2 The Use of Maximally Localized Wannier Functions
4.3 Molecular Dynamics Simulations
4.3.1 Introduction to MD
4.3.2 Ensembles: Thermostats and Barostats
4.3.3 Practical Considerations
4.4 Bottom-up Approach for Deriving Classical Potentials
4.4.1 Introduction
4.4.2 Describing Atomic Interactions Within a Classical Framework
4.4.3 The Procedure
4.5 Results
4.5.1 Polarizabilities of Ions in Solution
4.5.2 A New Force-Field for Ions in Solution
4.6 Conclusions
5 Implicit Solvent Molecular Descr.
5.1 Introduction
5.2 McMillan-Mayer Ion-Ion Potentials
5.2.1 Computing the Effective Interactions
5.2.2 Short-Range Solvent Averaged Interactions
5.2.3 Ion Association
5.3 Results
5.3.1 Simple Electrolytes
5.3.2 Highly Charged Asymmetric Electrolytes
5.4 Conclusions
6 From Molecular Descr. to Primitive Models
6.1 Introduction
6.2 Deriving the Simplest Implicit Solvent Model
6.3 Choosing the Reference System
6.3.1 Singular Reference Potentials
6.3.2 The Three – Component Model
6.4 The Free Energy of the Paired System
6.5 The Effective Interactions of the Paired System
6.5.1 The Pair-Ion Potential
6.5.2 The Pair-Pair Potential
6.5.3 Summary
6.6 The Structure of the Paired System
6.7 The Minimization Procedure
6.8 Results
6.9 Conclusions
7 Towards a Non-Additive Primitive Model
7.1 Motivation
7.2 Definitions
7.3 Second-Order Perturbation Theory
7.4 Ensemble Transformation
7.4.1 Basic Properties
7.4.2 Free Energy Derivatives
7.4.3 Grand-Potential Derivatives
7.5 Case Study: A Two Component System
7.5.1 Model
7.5.2 Functional Expansion
7.5.3 Diagrammatic Representation
7.6 Conclusions
8 Towards a Simple Theory of the Viscosity
8.1 Introduction
8.2 Mori-Zwanzig Projector Operator Formalism
8.3 Mode – Coupling Theory for the Viscosity
8.4 The Procedure
8.4.1 Calculation of the Binary Term
8.4.2 Calculation of the Mode-Coupling Term
8.5 Results
8.5.1 Molecular Dynamics Simulations
8.5.2 Mode-Coupling Calculations
8.6 Conclusions
9 General Conclusions
A Principles of Monte Carlo Simulations
B Averages and Error Calculations
B.1 Block-Averages
B.2 Real-time Updating During a Simulation
C Numerical Integration
D Numerical Laplace Inversion
D.1 Introduction
D.2 The Fourier Expansion
D.3 The Gaver Functional Expansion
E Monte Carlo Results
E.1 Implict vs Explicit Solvent (MC vs MD)
E.2 McMillan-Mayer Energy and Pressure
E.3 Structure of the Solute Gas
E.4 Minimum Distance Distributions
F PFT Results
F.1 Free Energy
F.2 Minimization Diameters